Virtual Signed Euler characteristics

Yunfeng Jiang Kansas University Richard Thomas Imperial College London

mathscidoc:1907.01002

Journal of Algebraic Geometry, 26, (2), 379-397, 2017
Roughly speaking, to any space M with perfect obstruc- tion theory we associate a space N with symmetric perfect obstruction theory. It is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (−1)-shifted cotangent bundle N. By localising from N to its C∗-fixed locus M this gives five notions of virtual signed Euler characteristic of M: (1) TheCiocan-Fontanine-Kapranov/Fantechi-G ̈ottschesignedvirtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande’s virtual Atiyah-Bott localisation of the virtual cycle of N to M, (3) Behrend’s Kai-weighted Euler characteristic localisation of the vir- tual cycle of N to M, (4) Kiem-Li’s cosection localisation of the virtual cycle of N to M, (5) (−1)vd times by the topological Euler characteristic of M. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.
virtual Euler number, symmetric obstruction theory
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@inproceedings{yunfeng2017virtual,
  title={Virtual Signed Euler characteristics},
  author={Yunfeng Jiang, and Richard Thomas},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190702174546710929392},
  booktitle={Journal of Algebraic Geometry},
  volume={26},
  number={2},
  pages={379-397},
  year={2017},
}
Yunfeng Jiang, and Richard Thomas. Virtual Signed Euler characteristics. 2017. Vol. 26. In Journal of Algebraic Geometry. pp.379-397. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190702174546710929392.
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